## Implied Content?

26 Oct

We have started our second unit on Geometric Relationships and Properties, focusing on the following learning objectives:

G-CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO 11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

I am not confident that my students have a better understanding of geometry than the students I have taught in the past. It feels like the students are missing big gaps of information. For example, I am hopeful that once I get students who have been through CCSS since kindergarten, we won’t have to stop our high school geometry course to define the base angles of an isosceles triangle. At least our students are learning to tell us when they don’t really understand a term so that we can add that to our running list of what we are assuming they know that they don’t.

I feel like students have missed out on some important content:

Isosceles Triangles – properties, relationships – We still need to prove the base angles theorem, but I usually leave this as an exercise on their unit assessment.

Triangle Inequality Theorem – It shows up in CCSS now for Grade 7, specifically in 7G.3:

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Am I the only one that it bothers that we are not naming the Triangle Inequality Theorem? I used the results of the Triangle Inequality Theorem quite a few times while studying mathematics. It seems like we would at least want to mention the name of it.

We included the Triangle Inequality Theorem as part of a bellringer this week, since it isn’t “official” content for our high school geometry course. Students explored possible values for the third side of a triangle, given that two of the side lengths were 5 and 7.

This was fine, but it felt rushed. I would have liked to spend more time on the Triangle Inequality Theorem. In fact, Triangle Inequality Theorem has always been a successful Geometry Nspired activity in which the students have determined that the sum of any two sides of the triangle must be greater than than the third.

What about inequalities in triangles? At what point in CCSS mathematics will students realize that the shortest side of a triangle is opposite the smallest angle? I put a question asking students to compare the side lengths of a triangle on yesterday’s bellringer, just to see if students already knew how to compare the lengths of sides in a triangle. They didn’t. But the diagram was interactive, and so we spent a few minutes discussing how to compare the angle measures given the side lengths.

It’s not like this is a difficult concept for students, but I like them to figure out the relationship between side lengths and angle measures; I don’t want to just tell them. It feels so unorganized for this question to be part of a bellringer instead of feeling like I can make it part of the lesson. If we approach this topic so that students have time to explore and understand, it takes 10-15 minutes of class.

What about angles in polygons? At what point are students going to learn how to determine the sum of the measures of the interior angles of a polygon? This was too much for a problem in a bellringer, so we have spent a day on it in class. We used an Interior Angles of Regular Polygons activity from Geometry Nspired. I loved watching the students work. They were attending to the practice of “look for regularity in repeated reasoning”. They were attending to the practice of “reason abstractly and quantitatively” as they generalized their results. We also used part of an Exterior Angles of Regular Polygons activity from Geometry Nspired, a great example of an activity that we used for about 5 minutes of whole class discussion.

The Math Nspired activities come with a student handout (in PDF and Word, so that you can easily edit before you give to your students), TNS document, and teacher notes (with Navigator tips, a few assessment questions, and possible solutions). They are all great resources, but I think it is freeing to remember that you don’t have to copy the student handout for every activity. Some activities lend themselves to a visual proof for students that can be discussed for a few minutes with the whole class.

As we move into the learning objective on parallelograms, I have so many questions. Do we forget about kites and trapezoids? In our deductive system, a trapezoid is not a parallelogram, although it is in other deductive systems. Or do we give our students any time to explore the properties of these special quadrilaterals?

Of course I understand that the standards help clarify what will be tested – and we can certainly teach additional content. But my fear is that I don’t have time for this additional content…and at least for now, what seems “additional” also seems necessary.

And so the journey continues ….